- Tests are graded:
- Curve: add 10 points to your percentage.
- I'll replace your worst midterm exam with the final, assuming the final is better.
- Homework, due Tuesday, 4/17:
Identify the knots (or links?) in this "story", which I call A Knotty Tale. You may need to apply the Reidemeister moves to convince yourself that a picture of a knot is really the unknot, say, but you don't need to tell me how you determined which knot or link each one is. Just put a name next to each one.
- What are the Reidemeister moves?
- Tools for showing that two projections of knots are really the same knot.
- Names of the moves reflect the number of strands involved. A Strand is a piece of a knot picture that goes from undercrossing to undercrossing, with only overcrossings inbetween.
- Reidemeister moves preserve the knot -- they don't change the knot.
-
Type I Type II Type III
- Using the Reidemeister moves
- to turn a projection of an unknot into another more complicated appearing unknot:
- Turning a more complicated appearing unknot into a simple circle unknot. (Here it is as a powerpoint presentation.)
- to turn a projection of an unknot into another more complicated appearing unknot:
- Reidemeister moves and Tricolorability
- Tricolorability:
"A knot is tricolorable if each strand of the
knot diagram can be colored one of three
colors, subject to the following rules:
- At least two colors must be used,
- at most three colors are used, and
- at each crossing, the three incident strands are either all the same color or all different colors.
- Tricolorability is preserved by Reidemeister moves:
Reidemeister Move I is tricolorable. Reidemeister Move II is tricolorable. Reidemeister Move III is tricolorable. 
- Therefore, if the projection of one knot is
tricolorable and the other isn't, they're different
knots (because we can't get from one to the other via
Reidemeister moves).
- Theorem: there are at least two different knots.
- The unknot is not tricolorable ("At least two colors must be used");
- The trefoil knot is tricolorable:
- The figure-eight knot is not tricolorable (it requires four colors):
- Danger: many knots are tricolorable --
being tricolorable doesn't mean that your knot
is the trefoil knot -- but it does mean that
your knot is not the unknot!
Links can be tricolorable, too -- for example, the unlink is tricolorable! (That's just two circles, one lying on top of the other, as in Borromean rings.)
- Tricolorability:
"A knot is tricolorable if each strand of the
knot diagram can be colored one of three
colors, subject to the following rules:
- Moebius bands are a part of geometry, but relate to a special area
called "topology".
Riddle: What's the difference between a donut and a coffee cup?
Answer: There isn't any! (to a topologist....)
- What is a Moebius band?
- Let's make some!
- How do we know that the moebius band is not "topologically equivalent" to a regular band?
- What happens when you cut a moebius band "in
half"? Let's check out a bug on a band....
- What if you cut it into "thirds"?
- By the way, did you know that there are two different
Moebius bands? They're mirror images of each other! (Symmetry, again....)
- Check yours against someone else's, and convince yourself that they're really different.
- Make one of each!
- My dad loved moebius bands so much....
- Applications of Moebius bands:
- Conveyor belts
- Kazakhstan's National Library in Astana
- Recycling Topology
- a history of the recycling symbol
- Recycling -- Moebius, or only Seemingly Moebius?
- A joke: "Why did the chicken cross the Mobius band?"
